To solve this problem, we need to simplify the given mathematical expression:
[tex]\[
\frac{6}{28x + 4} \div \frac{12}{35x + 5}
\][/tex]
### Step-by-Step Solution:
First, recall that division by a fraction is the same as multiplication by its reciprocal. We can rewrite the expression by multiplying with the reciprocal:
[tex]\[
\frac{6}{28x + 4} \times \frac{35x + 5}{12}
\][/tex]
Now, we handle the numerators and denominators separately. The numerator will now be the product of the original numerator of the first fraction and the numerator of the reciprocal of the second fraction:
[tex]\[
6 \times (35x+5)
\][/tex]
The denominator becomes the product of the original denominator of the first fraction and the denominator of the reciprocal of the second fraction:
[tex]\[
(28x+4) \times 12
\][/tex]
After performing these multiplications, we get:
[tex]\[
\frac{6(35x + 5)}{12(28x + 4)}
\][/tex]
We simplify this expression as follows:
1. Distribute the constants within the numerators and denominators:
[tex]\[
6 \times (35x + 5) = 210x + 30
\][/tex]
[tex]\[
12 \times (28x + 4) = 336x + 48
\][/tex]
2. The simplified expression is thus:
[tex]\[
\frac{210x + 30}{336x + 48}
\][/tex]
To confirm the simplification is correct, if we consider [tex]\( x = 7 \)[/tex] (as an example), we can substitute and verify:
### Substitute [tex]\( x = 7 \)[/tex] in both numerator and denominator:
- For the numerator:
[tex]\[
6 \times (35 \times 7 + 5) = 6 \times (245 + 5) = 6 \times 250 = 1500
\][/tex]
- For the denominator:
[tex]\[
12 \times (28 \times 7 + 4) = 12 \times (196 + 4) = 12 \times 200 = 2400
\][/tex]
This confirms that our final simplified fraction is:
[tex]\[
\frac{1500}{2400} = \frac{5}{8}
\][/tex]
Thus, the final simplified expression in terms of [tex]\( x \)[/tex] can be seen that our simplified form, once we cancel common factors, retains the structure matching:
[tex]\[
\boxed{\frac{5}{8}}
\][/tex]
Therefore, the correct answer is:
C. [tex]\(\frac{5}{8}\)[/tex]
